CK12 Calculus - Single Variable

• Learnhow to find the limit of indeterminateform by L’Hospital’s rule.

If the two functions and are both equalto zero at then the limit

cannotbe foundby directlysubstituting The reasonis becausewhenwe substitute the
substitutionwill produce knownas an indeterminateform, whichis a meaninglessexpression.
To workaroundthis problem,we use L’Hospital’s rule, whichenablesus to evaluatelimitsof indeterminate
forms.

L’Hospital’s RuleIf , and and exist,

where , then

The essenceof L’Hospital’s rule is to be able to replaceone limit problemwith a simplerone. In eachof the
examplesbelow, we will employthe followingthree-stepprocess:

1. Checkthat is an indeterminateform To do so, directlysubstitute into and
   If you get then you can use L’Hospital’s rule. Otherwise,it cannotbe used.


2. Differentiate and separately.


3. Find If the limit is finite,then it is equalto the originallimit.
   Example1:

Find
Solution:

When is substituted,you will get
ThereforeL’Hospital’s rule applies: